Procedure error

[PerlaC497]

If (xy)^1/2 = xy , what is the value of x + y?

(1) x = -1/2
(2) y is not equal to zero

Can you tell me what is wrong with my reasoning?
1) x= -1/2. If I plug that into the given equation, then:

(-1/2y)^1/2 = -1/2y which means that y must be negative, or else this is not possible.

Squaring both sides, then:
-1/2y = 1/4y^2
Dividing both sides by y, then:
-1/2 = 1/4y
Multiplying both sides by 4, then:
y= -2

SUFF.

[Sage Pearce-Higgins]

Be careful when dividing an equation by a variable! When you move from this equation: -1/2y = 1/4y^2 to this one: -1/2 = 1/4y, you have to consider that y could be zero. This gives two possible answers to the question 'What is x + y?", making the statement insufficient.